Performing remote power amplifier linearization

ABSTRACT

Estimating non-linearity of a power amplifier includes receiving signals at a first location. The signals include an input signal, a pre-distorted signal, and an output signal. The output signal exhibits distortion, including a non-linearity effect, with respect to the input signal. The power amplifier is located at a second location remote from the first location. The non-linearity of the power amplifier is estimated in accordance with the signals at the first location using an inverse model. Pre-distortion information is calculated according to the estimated non-linearity using the inverse model. The pre-distortion information is sent to a pre-distorter at the second location, where the pre-distorter can reduce the non-linearity effect using the pre-distortion information.

TECHNICAL FIELD

This invention relates generally to the field of power amplifiers andmore specifically to performing remote power amplifier linearization.

BACKGROUND

Power amplifiers are typically a major source of non-linearity incommunication systems. Tactical satellite transmission systems may beparticularly sensitive to non-linearity because of their inherently highpeak-to-average power ratios (PAPR). To reduce the effects ofnon-linearity, the non-linearity of a power amplifier can be estimated,and a signal can be pre-distorted in accordance to the estimate tocompensate for the non-linearity. Known techniques, however, may notsufficiently reduce the effects of non-linearity in certain situations.It is generally desirable to sufficiently reduce the effects ofnon-linearity.

SUMMARY OF THE DISCLOSURE

According to one embodiment of the present invention, estimatingnon-linearity of a power amplifier includes receiving signals at a firstlocation. The signals include an input signal, a pre-distorted signal,and an output signal. The output signal exhibits distortion, including anon-linearity effect, with respect to the input signal. The poweramplifier is located at a second location remote from the firstlocation. The non-linearity of the power amplifier is estimated inaccordance with the signals at the first location using an inversemodel. Pre-distortion information is calculated according to theestimated non-linearity using the inverse model. The pre-distortioninformation is sent to a pre-distorter at the second location, where thepre-distorter can reduce the non-linearity effect using thepre-distortion information.

Certain embodiments of the invention may provide one or more technicaladvantages. A technical advantage of one embodiment may be that off-linecomponents for determining non-linearity of on-line components may belocated remote from the on-line components. Locating the off-linecomponents remote from the on-line components may allow for remotelinearization of the on-line components while reducing power and spacerequirements for the on-line components.

Certain embodiments of the invention may include none, some, or all ofthe above technical advantages. One or more other technical advantagesmay be readily apparent to one skilled in the art from the figures,descriptions, and claims included herein.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention and itsfeatures and advantages, reference is now made to the followingdescription, taken in conjunction with the accompanying drawings, inwhich:

FIG. 1 is a block diagram of one embodiment of a system for linearizinga power amplifier;

FIG. 2 is a block diagram of one embodiment of a system that includes afilter smoother that performs a smoothing operation on pre-distorterweights before feeding the weights to an on-line pre-distorter; and

FIG. 3 is a flowchart illustrating one example of a method forestimating the non-linearity and memory depth of a power amplifier thatmay be used with the system of FIG. 1.

DETAILED DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention and its advantages are bestunderstood by referring to FIGS. 1 through 3 of the drawings, likenumerals being used for like and corresponding parts of the variousdrawings.

FIG. 1 is a block diagram of one embodiment of a system 10 forlinearizing a power amplifier. System 10 estimates the memory depth andnon-linearity of a power amplifier from forward and inverse transferfunctions, respectively. System 10 calculates one or morepre-distortions from the estimates, and then applies the pre-distortionsto compensate for the non-linearity in accordance with the memory depth.

According to the illustrated embodiment, system 10 includes one or moreon-line components 20, one or more off-line components 22, and a systemmonitor 24 coupled as shown. On-line components 20 perform on-lineprocessing. On-line processing may refer to signal processing that maybe performed in real time at approximately the same physical locationwhere the signal originates. On-line components 20 include componentsthat amplify an input signal x(n) to yield an output signal y(t), whilereducing or removing distortions of the output signal y(t). Off-linecomponents 22 perform off-line processing. Off-line processing may referto signal processing that may be performed in non-real time and notnecessarily at the same physical location where the signal originated.Off-line components 22 include components that are used to determineinformation used to reduce or remove the distortions of the outputsignal y(t). System monitor 24 sends and receives information to a userinterface to allow for controlling and monitoring of system 10.

On-line components 20, off-line components 22, and system monitor 24,may be located at one location or may be located at different locations.According to one embodiment, on-line components 20 and off-linecomponents 22 are located in the same device. According to anotherembodiment, on-line components 20 are located at a first location, whileoff-line components 22 and system monitor 24 are located at a secondlocation distant from the first location. The first location may be anysuitable distance from the second location, for example, more than oneinch, more than one yard, or more than one, ten, one hundred, or onemillion miles away from the second location.

The first location may refer to a site where signal amplification isneeded, but which might not be easily accessed to monitor or control thesignal amplification or which might not have sufficient resources suchas power or space to monitor or control the signal amplification.Examples of the first location include a spacecraft such as a satelliteor a space probe or a geographically remote place such as a cellularbase station. The second location may refer to a site that may be easilyaccessed or may have sufficient resources to monitor or control poweramplification. Examples of the second location include a space commandstation or a central office. On-line components 20 and off-linecomponents 22 may communicate signals to each other using any suitablecombination of transmitters and receivers. According to one embodiment,off-line components 22 may also be used to test power amplifiers in amanufacturing facility. The power amplifiers may be at one location ofthe facility, while the off-line components 22 may be at anotherlocation.

On-line components 20 receive input signal x(n), pre-distort inputsignal x(n) to yield pre-distorted signal x_(PD)(n). Input signaldescribes a pre-amplified signal, which may be filtered, converted, orotherwise processed before reaching the power amplifier. Output signaldescribes a signal that has been amplified by the distorting poweramplifier.

According to the illustrated embodiment, on-line components 20 includean impairment compensator 30, a pre-distorter 32, a switch 34, adigital-to-analog (D/A) converter and an intermediate frequency-to-radiofrequency (IF/RF) converter 38, a power amplifier (PA) 40, a switch 42,and a high power signal coupler 44 coupled as shown. In general,features of power amplifier 40 such as non-linearity and memory depthmay distort a signal. In order to reduce or remove the distortion,pre-distorter 32 may be used to introduce a pre-distortion into thesignal to compensate for the distortion.

Impairment compensator 30 operates on input signal x(n) to compensatefor impairments. An impairment may refer to any feature of components,other than power amplifier 40, that distorts a signal. An impairmenteffect may refer to any type of distortion such as a non-uniform delayor a gain or phase variation resulting from an impairment. For example,D/A converter 38 yields an amplitude roll-off factor. Impairmentcompensator 30 may receive compensation information for compensating forimpairment from off-line components 22 while power amplifier 40 is notamplifying. The off-line obtained compensation information may includecompensator coefficients, or weights, that may be used to adjust theoperating parameters of impairment compensator 30 to compensate forimpairments. According to another embodiment, impairment compensator 30and pre-distorter 32 may be combined into one module, yet configuredindependently to correct for non-linearity and impairments.

Pre-distorter 32 modulates input signal x(n) to introducepre-distortions into the signal to compensate for the non-linearity ofpower amplifier 40. Non-linearity may describe a feature of poweramplifier 40 that distorts a signal. A non-linearity effect may describeany type of distortion resulting from a non-linearity. Pre-distorter 32may receive pre-distortion information for compensating fornon-linearity from off-line components 22. Pre-distortion informationmay include pre-distorter coefficients, or weights, that may be used toadjust the operating parameters of pre-distorter 32. Typically, theoperation of a pre-distorter may be described by a set of poweramplifier inverse polynomials, where each polynomial applies to a tap ofthe pre-distorter that is implemented during a tape-delay line filter.The coefficients of the polynomials may be used to control the operatingparameters of the pre-distorter.

The operating parameters may be adjusted such that pre-distorter 32 hasthe inverse characteristics of power amplifier 40 to compensate for thenon-linearity of power amplifier 40. The design of pre-distorter 32 maybe regarded as a copy of the design of a power amplifier transferfunction obtained by an inverse model module 54 of offline components22, except that operating parameters may be used to make pre-distorter32 a complement, or inverse, of power amplifier 40. Switch 34 may beused to bypass pre-distorter 32 during initial assessments or during theestimation of the forward path impairments using a forward model module52 of offline components 22.

According to one embodiment, prior to processing by converters 38,pre-distorted signal x_(PD)(n) may be fed to an interpolator to increasethe sampling rate and reduce desired spectrum separation. Interpolatedpre-distorted signal x_(PD)(n) may be up-converted to a digitalintermediate frequency using a digital up-converter (DUC). According toone example, pre-distorted signal x_(PD)(n) may be over-sampled to arate that is exactly one-fourth of the frequency of the desiredintermediate frequency (IF) signal in order to perform digitalup-conversion without use of a dedicated numerically controlledoscillator (NCO) and real time multiplication.

D/A converter 38 converts pre-distorted signal x_(PD)(n), up-convertedfrom baseband to IF, from a digital format to an analog format. IF/RFconverter 38 converts pre-distorted signal x_(PD)(n) from anintermediate frequency to a radio frequency. Converters 38 may alsoapply a gain G_(IF) to pre-distorted signal x_(PD)(n).

Power amplifier 40 applies a power amplifier gain G_(PA) topre-distorted signal x_(PD)(n), and may apply distortions to thepre-distorted signal resulting from features such as non-linearity.Coupler 44 splits output signal y(n) into an output signal and afeedback signal. Switch 42 is used to bypass power amplifier 40 todetermine impairments introduced by other components of components 20,such as those caused by converters 38.

Off-line components 22 determine adjustments of on-line components 20that may be made to reduce distortion of output signal y(n). Off-linecomponents 22 include an analog-to-digital (A/D) converter and a radiofrequency-to-baseband (RF/BB) converter 50, a forward model module 52,an inverse model module 54, a multiplier 56, a summer 58, apost-processor 60, and a switch 62 coupled as shown. According to oneembodiment, off-line components 22 may be implemented as non-lineartape-delay-line (NTDL) complex filters known in the art.

According to the embodiment, off-line components 22 receive input signalx(n), pre-distorted signal x_(PD)(n), and output signal y(t) fromon-line components 20. A/D converter 50 converts the feedback signaly(t) from an analog format. RF/BB converter 50 converts signal y(n) fromIF or RF to baseband centered at, for example, zero Hz, using digitalsignal processing. Converters 50 may also apply a feedback gain G_(fb).

According to one embodiment, the feedback signal y(t) comprises anultrahigh frequency (UHF) carrier with a bandwidth equal toapproximately five to seven times the bandwidth of baseband signal x(n).For example, if the baseband signal x(n) has a bandwidth of ±15 MHz, thebandwidth of the feedback signal is ±75 MHz to ±105 MHz, which coversthe inter-modulations created up to the fifth and seventh ordernon-linearity inter-modulation products. Feedback signal y(t) may besampled directly at an ultrahigh frequency without sampling anintermediate frequency. For example, an ultrahigh frequency signalcentered at 255 MHz may be sampled at a rate of 150 MHz to yield adigital intermediate frequency signal of 45 MHz. The signal may beconverted to baseband so modules 52 and 54 may operate on the signal.According to another embodiment, the pre-distortions may be applied tothe IF signal, and not the baseband signal x(n). The feedback signaly(n) is translated to the same frequency as the pre-distorter IFfrequency.

Off-line components 22 may use blocks of samples of the on-line signals,so off-line components 22 may be operated at a rate different from thatof on-line components 20. For example, converters 50 may sample at arate of 150 MHz to capture 1024 samples, and then translate the samplesto baseband according to Equation (1): $\begin{matrix}\left. {y(n)}\leftarrow{{y(n)}\quad{\mathbb{e}}^{{- j}\quad\frac{2\pi\quad 45 \times 10^{6}\quad{({n = {0:1023}})}}{150 \times 10^{6}}}} \right. & (1)\end{matrix}$where n is the sample index. Before passing the signal to the modelmodules 52 and 54, the samples may be filtered to reduce or remove DCoffset, local oscillator leakage, radio frequency image, other feature,or any combination of the preceding.

Using the captured samples y(n), forward model module 52 and inversemodel module 54 generate forward and inverse models, respectively, ofpower amplifier 40. According to one embodiment, if pre-distorter 32 isbypassed, the current output signal y(n) may be modeled as the summationof a number of non-linear polynomials operating on the current sample aswell as past samples that mimic the temporal storage of energy due to PAmemory depth, which may be given by Equation (2): $\begin{matrix}{{\hat{y}(n)} = {\sum\limits_{m = 0}^{m = M}{{x\left( {n - m} \right)}\quad{\sum\limits_{k = 1}^{k = P}{b_{mk}{{x\left( {n - m} \right)}}^{k - 1}}}}}} & (2)\end{matrix}$where b_(mk) are complex coefficients that model the PA non-lineardynamics, x(n) is the baseband input of power amplifier 40, n is thesample index, m is the sample delay representing the memory index, and Mis the maximum value of delays that represent the memory depth. Forpre-distorter 32 to effectively pre-distort a signal, typically M isneeded. Forward model module 52 estimates memory depth M, which inversemodel module 54 uses to estimate coefficients for pre-distorter 32.

Forward model module 52 applies a forward transfer function of a forwardmodel to signals x(n) and ŷ(n) to estimate the memory depth M of poweramplifier 40 and to generate information for system monitor 24. Aforward model represents the forward dynamics of power amplifier 40 asdescribed by a forward transfer function that estimates the equivalentbaseband linear and non-linear response using Equation (2). The memorydepth M describes the number delay taps that may be used to model poweramplifier 40.

Inverse model module 54 applies an inverse transfer function of aninverse model to signals x_(PD)(n) and y(n) to determine pre-distortioninformation such as pre-distorter weights that pre-distorter 32 may useto pre-distort the signal. An inverse model represents the inversetransfer function of power amplifier 40 as described by an inversetransfer function. The function may be programmed into pre-distorter 32to be provided in series to power amplifier 40 to reduce thenon-linearity of power amplifier 40. According to one embodiment, thepre-distortion information may be generated in accordance with memoryinformation provided by module 52 in order to account for the memorydepth of power amplifier 40. Inverse model module 54 may also generatecompensation information to compensate for impairments introduced by thecomponents of system 10 using the inputs x(n) and x_(PD)(n).

Inverse model module 54 reduces or removes the PA forward phase shift,so the complex multiplier 56 may be used to multiply feedback signaly(n) by a desired complex gain G_(desired)e^(jθ) ^(desired) , whichintroduces a desired phase θ_(desired) and gain scaling G_(desired) intopre-distorter 32, and j is complex notation corresponding to √{squareroot over (−1)}. The feedback signal y(n) may be multiplied by thecomplex gain y(n)←y(n)G_(desired)e^(jθ) ^(desired) in order to adjustthe total overall forward gain and phase of power amplifier 40,discussed below in more detail. Post-processor 60 may be used to performpost-processing on the pre-distortion information such as pre-distorterweights. Post-processing may make system 10 more robust against suddeninput signal variations or any other unexpected changes in the outputsignal y(n).

System monitor 24 communicates information to, from, or both to and froma user interface to allow for controlling and monitoring of system 10 bya user. Information provided to a user interface may include theamplitude-to-amplitude (AM/AM) and amplitude-to-phase (AM/PM) distortionof the signals, the temporal alignment of the PA input and outputbaseband signals, the correlation function indicating the alignment ofthe baseband signals, other information, or any combination of thepreceding. Information received from a user interface may includesetting for an operating power level, a desired power amplifier gain, ora total power amplifier phase shift, other information, or anycombination of the preceding. According to one embodiment, systemmonitor 24 may be used to control the gain and phase of power amplifier40 as discussed below.

Modifications, additions, or omissions may be made to system 10 withoutdeparting from the scope of the invention. For example, post-processor60 may be omitted. Moreover, fewer, or other modules may perform theoperations of system 10. For example, the operations of system monitor24 may be performed by more than one module. Additionally, operations ofsystem 10 may be performed using any suitable logic comprising software,hardware, other logic, or any suitable combination of the preceding. Asused in this document, “each” refers to each member of a set or eachmember of a subset of a set.

FIG. 2 is a block diagram of one embodiment of a system 110 thatincludes a filter smoother 172 that performs a smoothing operation onpre-distorter weights before feeding the weights to an on-linepre-distorter 132. According to one embodiment, system 110 includespre-distorter 132, a digital-to-analog (D/A) converter and anintermediate frequency-to radio frequency (I/F-R/F) converter 138, apower amplifier 140, an analog-to-digital (A/D) converter and a radiofrequency-to-baseband frequency (RF/BB) converter 150, an inverse modelmodule 154, a summer 170, and a filter smoother 172 coupled as shown.According to the illustrated embodiment, pre-distorter 132, converters138, power amplifier 140, and converters 150 may operate in a mannersubstantially similar to pre-distorter 32, converters 38, poweramplifier 40, and converters 50 respectively of system 10 of FIG. 1.

The post-processing module comprising filter smoother 172 may beimplemented as a first order filter smoother that performs a first orderrecursive operation. Filter smoother 172 performs a smoothing operationon pre-distorter weights obtained by off-line estimator 174 beforefeeding the weights to pre-distorter 132. Filter smoother 172 includes,a multiplier 176, a summer 178, a delay 180, and a multiplier 182coupled as shown. The post-processing module may alternatively comprisea moving average filter, a Savitsky-Golay smoothing filter, or othersuitable filter.

As an example, samples sets of signals x_(PD)(n) and {circumflex over(x)}_(PD)(n) are processed. A sample set of a signal may include anysuitable number of samples from the signal, for example, 1024 samples.According to the illustrated embodiment, delay 168 delays signalx_(PD)(n) such that signal {circumflex over (x)}_(PD)(n) arrives atsummer 170 at substantially the same time as the corresponding signalx_(PD)(n) of the same sample. Summer 170 determines the differencebetween signals x(n) and {circumflex over (x)}(n) to generate errorsignal e_(inv)(n)=x_(PD)(n)−{circumflex over (x)}_(PD)(n). Error signale_(inv)(n) and delayed samples of x_(PD)(n) are passed to estimator 174,which adjusts the weights of off-line processor 154 in order to minimizethe next error signal e_(inv)(n+1).

Estimator 174 determines if error signal e_(inv)(n) satisfies an errorthreshold, indicating that the weights are acceptable. Error signale_(inv)(n) may be measured using the normalized mean square error(NMSE), which is the logarithmic norm power that may be given byEquation (3): $\begin{matrix}{{NMSE} = {10\quad\log_{10}\frac{{{{\sum\limits_{n}{x_{PD}(n)}} - {{\hat{x}}_{PD}(n)}}}^{2}}{\sum\limits_{n}{{x_{PD}(n)}}^{2}}}} & (3)\end{matrix}$In this example, error signal e_(inv)(n) satisfies the error thresholdif the NMSE is less than −40 dB. If error signal e_(inv)(n) satisfiesthe error threshold, estimator 174 sends weights ŵ_(k) of inverse modelmodule 154 to complex multiplier 176, where bold text represents avector or matrix. According to one example, estimator 174 may send onlyselected weights to multiplier 176. For example, the weightscorresponding to the n=974 sample of a set of 1024 samples of x_(PD)(n)may be selected as the kth optimal weights. The weights corresponding tosamples after n=1024 and before n=974 may be discarded to protectagainst end transients. The weights at n-974 may be validated andchecked to make sure they do not exceed the allowable range ofpre-distorter 32.

Multiplier 176 multiplies weights ŵ_(k) by μ, where k is the index ofthe weight updates and μ represents a smoothing factor, 0≦μ≦1. Thepost-processing smoothing factor represents the desired degree ofsmoothing. Delay 180 delays the output from multiplier 178, andmultiplier 182 multiplies the output by 1-μ. Summer 178 adds the outputfrom multipliers 176 and 178 to yield smoothed weights {tilde over(w)}_(k). In this embodiment, an “iteration” may refer to a cycle wherea sample or weight is processed and passed on to on-line pre-distorter132.

The smoothing operation may be described by Equation (4):$\begin{matrix}{{\overset{\sim}{w}}_{k} = \left\{ \begin{matrix}{{\hat{w}}_{k},} & {k < L} \\{{{\mu\quad{\hat{w}}_{k}} + {\left( {1 - \mu} \right)\quad{\overset{\sim}{w}}_{k - 1}}},} & {k > L}\end{matrix} \right.} & (4)\end{matrix}$where k represents the iteration index, L represents the iteration atwhich smoothing begins that is supplied by system monitor 24 of FIG. 1,and ŵ_(k) represents the set of selected weights after k iterations frominverse model 174.

The robustness of this scheme is evident from the variance and noisereduction of coefficients {tilde over (w)}_(k) compared to that ofŵ_(k). The variance and noise reduction due to the smoothing operationis given by $\frac{\mu}{2 - \mu}.$The smoothing operation however, may result in a slow adaptationresponse to reach the steady state values for the optimal weights. Forexample, if μ=0.4, the steady state is reached after ten iterations, andthe noise and variance reduction is 12 dB. If μ=0.2, the steady state isreached after twenty iterations, and the noise and variance reduction is19 dB. The value of L in Equation (3) may be selected such that thesmoothing operation does not start until system 10 is into goodconvergence areas, which may be at L=5 to L=10 iterations.

Modifications, additions, or omissions may be made to system 110 withoutdeparting from the scope of the invention. Moreover, the operations ofsystem 110 may be performed by more, fewer, or other modules.Additionally, operations of system 110 may be performed using anysuitable logic comprising software, hardware, other logic, or anysuitable combination of the preceding.

FIG. 3 is a flowchart illustrating one example of a method forestimating the non-linearity of a power amplifier that may be used withsystem 10 of FIG. 1. In the example method, system 10 pre-processessignals and scales the gain of the components of system 10. System 10estimates and compensates for the impairments of system 10, and thenestimates and compensates for the non-linearity of power amplifier 40 inaccordance with the memory depth of power amplifier 40.

Steps 200 and 202: Initial Steps

The method begins at step 200, where sampled signals are received atoff-line components 22. The strips of sampled signals include inputsignal x(n), pre-distorted signal x_(PD)(n), and output signal y(n).Pre-processing procedures are performed on the sampled signals at step202. According to one embodiment, off-line components 22 maydown-convert signals y(n) from an intermediate frequency to a basebandfrequency, and may filter signals y(n) to reduce or remove the effectsof DC offsets, local oscillator leakage, harmonics problems, spectralimages, other problem, or any combination of the preceding. The time lagoffset between the input sampled signal x(n) and the output sampledsignal y(n) may also be reduced or removed.

Step 204: Scaling Forward Path Gain

The gain values for the components of system 10 are scaled at step 204.The scaling may be performed off-line and even remotely from a distanceusing the information supplied by system monitor 24 and samples x(n) andx_(PD)(n). The components include power amplifier 40, D/A and IF/RFconverter 38, and A/D and RF/BB converter 50. The pre-distorted outputbaseband samples may be given by Equation (5): $\begin{matrix}{{x_{PD}(n)} = {{{x(n)}a_{01}} + {\sum\limits_{k = 2}^{k = p}{a_{0k}{{x(n)}}^{k - 1}}} + {{x\left( {n - 1} \right)}a_{11}} + {\sum\limits_{k = 2}^{k = p}{a_{1k}{{x\left( {n - 1} \right)}}^{k - 1}}} + {{x\left( {n - 2} \right)}a_{21}} + {\sum\limits_{k = 2}^{k = p}{a_{2k}{{x\left( {n - 2} \right)}}^{k - 1}}} + {\ldots\quad{\sum\limits_{m = 0}^{m = M}{{x\left( {n - m} \right)}\quad{\sum\limits_{k = 1}^{k = P}{a_{mk}{{x\left( {n - m} \right)}}^{k - 1}}}}}}}} & (5)\end{matrix}$where a_(mk) represent pre-distorter weights, M represents memory depth,and P represents the polynomial order of the pre-distorter polynomial.Pre-distorter weights a_(mk) make the pre-distorter transfer functionsubstantially equivalent to the inverse transfer function of poweramplifier 40. Off-line estimator 54 may be used to determine theweights. Off-line amplifier 40 forward model estimator 52 may supplyestimates of memory depth M and polynomial order P. As an example, ifM=2, the linear terms of the first three taps may be given by Equation(6):x _(PD)(n)=x(n)â ₀₁ +x(n−1)â ₁₁ +x(n−2)â ₂₁   (6)

where â₀₁, â₁₁, and â₂₁ are the estimates of the first three linear tapterms obtained from the off-line inverse model that estimates therequired actual coefficients a₀₁, a₁₁, and a₂₁.

During initiation of the off-line estimation, the forward path gain maybe set to provide the overall gain for the PA drive to reach a desiredoutput peak power. The gain may be set by using${\frac{x_{PD}(n)}{x(n)} = a_{01}},$where the other coefficients are set to zero. The total forward pathgain may be given by Equation (7): $\begin{matrix}{\frac{y(n)}{x(n)} = {{a_{01}}\quad G_{IF}G_{PA}}} & (7)\end{matrix}$

Initially, assuming that the input samples are |x(n)|≦1, pre-distorterweights a_(mk), except linear term a₀₁, may be set to zero.Pre-distorted signal x_(PD)(n) may then be given by Equation (8):x _(PD)(n)=x(n)a ₀₁   (8)The power amplifier output peak power P_(out)(dBm) may then be given byEquation (9): $\begin{matrix}{{P_{out}\quad({dbm})} = {{10\quad{\log_{10}\left( {\frac{1}{N}\quad{\sum\limits_{1}^{N}{{x(n)}}^{2}}} \right)}} + {PAPR} + {G_{fwd\_ tot}\quad\left\lbrack {{dB}\quad m} \right\rbrack}}} & (9)\end{matrix}$where N represents a statistically sufficient number of samples in asample set, PAPR represents the peak-to-average power ratio of thebaseband signal in dB, and G_(fwd) _(—) _(tot) represents the totalforward path gain.

The total forward path gain may take into account gain from componentssuch as converters 38, digital up-converter, and power amplifier 40,where the gain of the impairment compensator 30 is assumed to be one orzero dB, and small signal gain. The ratio PAPR may be given by Equation(10): $\begin{matrix}{{PAPR} = {\frac{\max\left( {{x(n)}}^{2} \right)}{\frac{1}{N}\quad{\sum\limits_{1}^{N}{{x(n)}}^{2}}}\quad\lbrack{dB}\rbrack}} & (10)\end{matrix}$and the total forward path gain G_(fwd) _(—) _(tot) may be given byEquation (11):G _(fwd) _(—) _(tot)=(|a ₀₁ |+G _(IF) +G _(PA)) [dB]  (11)where G_(IF) represents the combined digital up-converter gain, D/Aconverter sensitivity gain, and IF/RF converter gain, and G_(PA)represents the gain of power amplifier 40.

The magnitude of the initial setting for the pre-distorter linear terma₀₁ may be chosen to achieve a desired power amplifier output peak powerP_(out) and may be given by Equation (12):|a ₀₁ |=P _(out)(dbm)−G _(IF) −G _(PA) [dB]  (12)Pre-distorter linear term |a₀₁| may be converted from a power dB settingto a voltage dB setting to yield actual linear gain$G_{PD} = 10^{(\frac{a_{01}}{20})}$that may be programmed into a field programmable gate array (FPGA)implementation of pre-distorter 32.

When the off-line inverse model converges, the gain of pre-distorter 32is not defined by the linear term of just the first tap, but may begiven by${{\sum\limits_{m = 0}^{m - M}{\sum\limits_{k = 1}^{k = P}a_{mk}}}}.$For example, if the off-line forward path estimate yields memory depthM=3 and polynomial order P=3, the forward gain due to the pre-distorteris readjusted from $\frac{x_{PD}(n)}{x(n)} = a_{01}$to Equations (13): $\begin{matrix}\begin{matrix}{\frac{x_{PD}(n)}{x(n)} = {{\sum\limits_{m = 0}^{m = 2}{\sum\limits_{k = 1}^{k = 3}a_{mk}}}}} \\{= {{{\sum\limits_{k = 1}^{k = 3}a_{0k}} + {\sum\limits_{k = 1}^{k = 3}a_{1k}} + {\sum\limits_{k = 1}^{k = 3}a_{2k}} +}}} \\{= {{\underset{\underset{{tap} - 1}{︸}}{\left( {a_{01} + a_{02} + a_{03}} \right)} + \underset{\underset{{tap} - 2}{︸}}{\left( {a_{11} + a_{12} + a_{13}} \right)} + \underset{\underset{{tap} - 3}{︸}}{\left( {a_{21} + a_{22} + a_{23}} \right)}}}}\end{matrix} & (13)\end{matrix}$The closed loop system readjusts the pre-distorter gain to account forthe gain incorporated into the additional taps that are used to mimicthe memory response of power amplifier 40.

According to one embodiment, system monitor 24 may be used to change thegain and phase of power amplifier 40 by adjusting linear term a₀₁. Ifthe phase of pre-distorter 32 is initially programmed to be zerodegrees, where the gain is real, and if off-line components 22 introducenegligible phase shift, the estimated linear term a₀₁ has a phase shiftthat represents the total phase shift between the input and output ofpower amplifier 40. A calculated linear term a₀₁ may be used to apply aphase offset θ_(offset) to achieve a desired phase θ_(desired), whichmay be described by Equation (14):θ_(desired) =arg{a ₀₁ }=arg{a ₀₁}+θ_(offset)   (14)The gain and phase adjustment can be applied systematically as shown inFIG. 1 by incorporating a gain into the PA captured feedback signalgiven by y(n)←y(n)G_(desired)e^(jθ) ^(desired) , which may be appliedusing multiplier 56 to change the gain and phase of power amplifier 40.

The drive level setting for D/A converter 38 may be calculated using asample that has a unity maximum amplitude and is scaled by a₀₁. Thesample is transmitted and the power setting calculated such that theoutput of D/A converter 38 is reduced by, for example, 6 dB from itsfull scale capabilities, and the output is reduced to account for thePAPR. For example,${x_{PD}a_{01}} = {\frac{1}{2}{\left( {1 + {0j}} \right).}}$The resulting D/A gain G_(DAC) may be estimated from the measured outputpeak power according to Equation (15): $\begin{matrix}{G_{DAC} = {{P_{{out},{DAC}}\quad\left( {{dB}\quad m} \right)} - {10\quad{{\log_{10}\left( {\frac{1}{2}}^{2} \right)}\quad\lbrack{dB}\rbrack}}}} & (15)\end{matrix}$According to one embodiment, D/A converter 38 may be set with the worstloading case by considering the PAPR and the full-scale power handling(dB-FS). For example, if the setting of${x_{PD}a_{01}} = {\frac{1}{2}\left( {1 + {0j}} \right)}$yielded −15 dBm, the digital-to-analog converter gain is G_(DAC)=−15dB−(−6 dB)=−9 dB.

Scaling Feedback Path Gain

The feedback signal y(n) generated from output signal y(t) may be givenby Equation (16):y(n)=x(n)(G _(fwd) _(—) _(tot) G _(ADC) G _(RF) _(—) _(BB))   (16)where G_(ADC) represents the analog-to-digital converter sensitivitygain, G_(RF) _(—) _(BB) represents the total RF feedback gain, which mayinclude the gain from power amplifier output directional couplerattenuation, from any attenuator used to adjust the A/D input level, andfrom digital filtering and translation of the AID output digital samplesto baseband samples.

According to Equation (12), to compare output signal y(n) withpre-distorted signal x_(PD)(n), the feedback path may be scaled by afeedback gain value that equates the signal x_(PD)(n) from the output ofpre-distorter 32 to the signal {circumflex over (x)}_(PD)(n) at theoutput of inverse model module 54. The feedback gain may be written as${g_{FB} = 10^{(\frac{- {G_{FB}}}{20})}},$where g_(FB) may be given by Equation (17):G _(FB) =G _(ADC) +G _(RF) _(—) _(BB) +P _(out) [dB]  (17)

Step 210: Estimating and Reducing Impairment Effects

Impairment effects due to the forward path are reduced at step 210.Impairment effects may be reduced by estimating the impairments, whilepower amplifier 40 is bypassed by switch 42, using inverse model module54 to yield a set of impairment compensation weights, and applying theimpairment compensation weights to impairment compensator 30 to reduceimpairment effects. Inverse model module 54 may estimate impairments byclosing switch 42 to bypass power amplifier 40 and closing switch 34 ifpre-distorter 32 has been programmed. The transfer function of theforward path is then equalized in order to isolate the impairments.

According to one embodiment, the forward path may be equalized byassuming that the undesired gain and phase variations in the forwardpath can be modeled using a finite impulse response (FIR) filter with Ncoefficients. Inverse model module 54 and impairment compensator 30 maybe programmed to implement the FIR filter model. Module 54 inconjunction with minimum least-square error (MSE) reduction algorithmmay be used to find the coefficients to load into impairment compensator30.

The output of impairment compensator 30 may be given by Equation (18):x _(PD)(n)=x(n)   (18)Inverse model module 54 may be programmed with a filter to yield anoutput given by Equation (19): $\begin{matrix}{{{\hat{x}}_{PD}(n)} = {\sum\limits_{k = 0}^{N - 1}{b_{k}\quad{y\left( {n - k} \right)}}}} & (19)\end{matrix}$where b_(k) are the coefficients for inverse model module 54. Thecoefficients operate to reduce or cancel the gain and phase variationsof the forward path G_(IF)H_(IF) _(—) _(RF) from converters 38 to yieldlinear gain G_(IF). The coefficients b_(k) for impairment compensator 30are derived using inverse model module 54 in conjunction with MSEreduction algorithm described by Equation (20):min|e _(inv)|²=min|x _(PD) −{circumflex over (x)} _(PD)|²   (20)where x_(PD) is written as x_(PD)=(n).

The impairment of converters 38 may be reduced or removed as follows.The output of impairment FIR filter is initialized with a set of weightssuch that the x_(PD)=x. Impairment compensator 30 and inverse modelmodule 54 are simultaneously programmed with a set of weights thatreduces or cancels undesired gain and phase of converters 38, which maybe designated using a linear gain and a frequency dependent transferfunction given by H_(IF) _(—) _(RF)G_(IF). The impairment term H_(IF)_(—) _(RF) may be reduced or removed leaving the linear gain G_(IF).

The output of impairment compensator 30 may be given by Equation (21):x_(PD)=H_(on) _(—) _(l)x   (21)where H_(on) _(—l) is the on-line transfer function. Pre-distorter 32and power amplifier 40 are bypassed, so the digital baseband samplesfrom converters 38 may be given by Equation (22):y=z=H_(IF) _(—) _(RF)G_(IF)x_(PD)   (22)and the output of inverse model module 54 may be given by Equation (23):{circumflex over (x)}_(PD)=H_(off) _(—) _(l)y=H_(off) _(—) _(l)H_(IF)_(—) _(RF)G_(IF)x_(PD)   (23)

If there were no impairments from converters 38, the output of module 54would be equal to the input of converters 38, that is, x_(PD)=x_(PD).The error function that can adapt the off-line transfer function to bean exact inverse of the impairment is given by Equation (24):$\begin{matrix}{{e_{inv}(n)} = {{x_{PD} - {\hat{x}}_{PD}} = {\left\lbrack {{x_{PD}(n)} - {\sum\limits_{k = 0}^{N - 1}{b_{k}{y\left( {n - k} \right)}}}} \right\rbrack = 0}}} & (24)\end{matrix}$Substituting values for {circumflex over (x)}_(PD), the error functionis given by Equation (25):e _(inv) =x _(PD) −H _(off) _(—) _(l) H _(IF) _(—) _(RF) G _(IF) x_(PD)=0   (25)The error function reaches a minimum when the inverse model cancels theimpairment from converters 38, which may be expressed by Equations (26):$\begin{matrix}\begin{matrix}{{H_{off\_ l}H_{IF\_ RF}G_{IF}} = 1} \\{H_{off\_ l} = \frac{1}{H_{IF\_ RF}G_{IF}}}\end{matrix} & (26)\end{matrix}$

To reduce or cancel just the impairment but not the linear gain G_(IF),the coefficients may be scaled to keep the linear gain according toEquation (27): $\begin{matrix}{\left. H_{on\_ l}\leftarrow{H_{on\_ l}G_{IF}} \right. = \frac{1}{H_{IF\_ RF}}} & (27)\end{matrix}$Now that the forward path has only linear gain G_(IF), modules 52 and 54are initiated again to estimate the PA electrical memory depth andnon-linearity of power amplifier 40 without disturbances fromimpairments.

According to another embodiment, on-line impairment compensator 30 maybe omitted. Impairment distortion such as group delay and gainvariation, however, may prohibit optimal estimation and correction ofthe PA electrical memory and non-linearity. If the impairment inversetransfer function H_(IF) _(—) _(RF) ^(—1) is cascaded in the off-linefeedback path along with the inverse model function given by Equation(19), which effectively makes the off-line model transfer function(H_(IF) _(—) _(RF) _(—1) H_(off) _(—) _(l)), the error signal may begiven by Equation (28):e _(inv) =x _(PD) −{circumflex over (X)} _(PD)=0   (28)

The pre-distorted signal may be given by Equation (29):{circumflex over (x)} _(PD) =x _(PD) G _(IF) H _(IF) _(—) _(RF) G _(PA)H _(PA) G _(fb)(H _(IF) _(—) _(RF) H _(off) _(—) _(l))   (29)Signal distortion is described by the PA non-linear function H_(PA) andthe forward path undesired frequency dependent function H_(IF) _(—)_(RF). Substituting the values for the pre-distorted signals into theerror yields Equation (30):e _(inv) =x _(PD) −x _(PD) G _(IF) H _(IF) _(—) _(RF) G _(PA) H _(PA) G_(fb)(H _(IF) _(—) _(RF) H _(off) _(—) _(l))=1   (30)Cascading the forward impairment function with the off-line inversemodel yields H_(IF) _(—) _(RF)H_(IF) _(—) _(RF)=1, which simplifies theerror signal to Equation (31):G _(IF) H _(IF) _(—) _(RF) G _(PA) H _(PA) G _(fb)(H _(IF) _(—) _(RF) ⁻¹H _(off) _(—) _(l))=1   (31)where G_(IF)G_(PA)H_(PA)G_(fb)H_(off) _(—) _(l)=1.

The total feedback gain is set to be equal to the forward gain, that is,${G_{fb} = \frac{1}{G_{IF}G_{PA}}},$so the off-line model transfer function is given by Equation (32):$\begin{matrix}{H_{off\_ l} = \frac{1}{H_{PA}}} & (32)\end{matrix}$Accordingly, the off-line model is an inverse of power amplifier 40, andmay be used to reduce or cancel PA non-linearity without anyinterference from the forward path transfer function H_(IF) _(—) _(RF).

Step 212: Estimating the PA Electrical Memory Depth and Its ForwardTransfer Function

The memory depth of power amplifier 40 is estimated at step 212 togenerate memory information. The memory depth describes the number ofdelay taps that may be required to model power amplifier 40 to programpre-distorter 32 to reduce or cancel the PA non-linearities. A delay tapmay refer to a single sample delay given by ${\tau = \frac{1}{F_{s}}},$where F_(s) is the complex signal sampling frequency of pre-distorter32, that is, the time between two samples of x_(PD)(n).

If the pre-distorter 32 is bypassed, that is, x_(PD)(n)=x(n), the memoryinformation describes the memory depth of power amplifier 40, and mayinclude the memory constant of power amplifier 40 and the number of thedelay taps for modeling power amplifier 40. The amplifier memory andnon-linearity may be described by the PA output given by Equation (33):$\begin{matrix}{{y(n)} = {\sum\limits_{m = 0}^{m = M}{{x\left( {n - m} \right)}\quad{\sum\limits_{k = 1}^{k = P}{c_{mk}{{x\left( {n - m} \right)}}^{k - 1}}}}}} & (33)\end{matrix}$where c_(mk) are complex coefficients that describe the PA forwardtransfer function that includes memory and non-linearity, M is thememory depth, and P is the order of the polynomial that best fits the PAmeasurements, which describes the transfer function of AM/AM and AM/PMdistortions.

Forward model module 52 may estimate the memory depth using a memoryestimation procedure that involves two processes. First, forward modelmodule 52 determines the order of an output signal polynomialcorresponding to output signal y(n) that fits measured non-linearitydistortion, such as measured AM/AM and AM/PM, for a memory-less system.The order may be, for example, a fifth or seventh order. The selectedorder may be based on the best value of P such that the PA observedsamples y(n) and x(n) best fit Equation (34): $\begin{matrix}{{\hat{y}(n)} - {{x(n)}\quad{\sum\limits_{k = 1}^{k = P}{c_{k}\quad{{x(n)}}^{k - 1}}}}} & (34)\end{matrix}$where $\frac{\hat{y}(n)}{x(n)}$describes the AM/AM distortion, and tan$\tan^{- 1}\left( \frac{\hat{y}(n)}{x(n)} \right)$describes the AM/PM distortion.

Second, forward model module 52 performs a delay tap identificationprocess using the established polynomial order to determine the memorydepth. The number of delay taps M is first varied while the magnitude ofthe complex coefficients |c_(k)| of the output signal polynomial ismonitored with respect to a predetermined tap magnitude threshold. Thenumber of delay taps required to model power amplifier 40, that is, thememory depth, is the delay value at which the tap magnitude, or signalpower, decreases below a predetermined threshold.

The magnitude of the coefficients may be monitored in any suitablemanner. According to one embodiment, the magnitude of coefficients of anoutput signal polynomial describing output signal y(t) for an impulseinput signal x(t)=δ(t), where${\delta\left( {t - \tau} \right)} = \left\{ {\begin{matrix}{1,} & {t = \tau} \\0 & {else}\end{matrix},} \right.$may be monitored. Using Equation (33), the memory activity signal y(t)may be described by Equation (35): $\begin{matrix}{{y(n)} = {{\sum\limits_{m = 0}^{m = M}{\sum\limits_{k = 1}^{k = P}{a_{mk}\quad{\delta\left( {n - m} \right)}{{\delta\left( {n - m} \right)}}^{k - 1}}}} = \left\{ \begin{matrix}{c_{mk},} & {n = m} \\0 & {else}\end{matrix} \right.}} & (35)\end{matrix}$

According to another embodiment, the magnitude of the first polynomialterm of each delay tap may be monitored. Equation (35) profiles each tapvalue magnitude. The first term of every linear term of a tap istypically the largest, so a threshold may be selected by counting thenumber of taps before the coefficient magnitude drops below the pointwhere the memory is assumed to be negligible, such as below 10 dB of thelinear term. According to one embodiment, the memory depth for 40 dBm ofoutput power may be up to seven delay taps. According to anotherembodiment, the memory depth for 45 dBm of output power may be up totwelve delay taps.

The value of the polynomial order P may be estimated by forward modelmodule 52 performing a minimum least square fit of the PA baseband datay(n) to a model given by Equation (36): $\begin{matrix}{e_{fwd} = {{\hat{y}(n)} - {{x(n)}\quad{\sum\limits_{k = 1}^{k = P}{c_{k}\quad{{x(n)}}^{k - 1}}}}}} & (36)\end{matrix}$An error function may be given by Equation (37): $\begin{matrix}{e_{fwd} = {{{y(n)} - {\hat{y}(n)}} = {{y(n)} - {{x(n)}\quad{\sum\limits_{k = 1}^{k = P}{c_{k}\quad{{x(n)}}^{k - 1}}}}}}} & (37)\end{matrix}$

A value of P such that $\min\limits_{p}{e_{fwd}}^{2}$may be determined.

Values of the forward model are substituted in Equation (37) asdescribed by Equation (38): $\begin{matrix}{e_{fwd} = {{{{x\left( \frac{1}{H_{IF\_ RF}} \right)}\left( {G_{IF}H_{IF\_ RF}} \right)\left( {G_{PA}H_{PA}} \right)\left( G_{fb} \right)} - {x\quad H_{fwd}}} = 0}} & (38)\end{matrix}$to yield Equation (39):(G _(IF))(G _(PA) H _(PA))(G _(fb))=H _(fwd)   (39)

By adjusting the total feedback gain to be equal to the combined forwardpath gain, which is described by Equation (40):$G_{fb} = \frac{1}{G_{IF}G_{PA}}$the forward model converges to an estimate of the PA forward path givenby Equation (41):H_(PA)=H_(fwd)   (41)which takes a forward non-linear polynomial function. The function maybe given by Equation (42): $\begin{matrix}{\frac{\hat{y}}{x} = {{\sum\limits_{k = 1}^{k = P}{c_{k}{x}^{k - 1}}} = {c_{1} + {c_{2}{x}} + {c_{3}{x}^{2}} + {c_{4}{x}^{3}} + \ldots}}} & (42)\end{matrix}$

The coefficients {c_(k)} may be estimated using least square fitting ofsamples of input x(n) and output y(n). These coefficients are passed tosystem monitor 24 which uses the coefficients to provide the monitorcontroller with power amplifier parameters such as signal gain, inputthird order intercept point (IIP3), and one-dB input power values(P_(in,1dB)). The IIP3 is given by${20\quad{\log_{10}\left( \frac{c_{1}}{c_{3}} \right)}},$while the P_(in,1dB) (dBm) is approximated by P_(in,1dB)=IIP3−9.63 1 dB[dBm].

Now that the PA is characterized in terms of P using Equation (33), thememory depth may be estimated next using off-line forward path model 52.Off-line forward model module 52 is programmed with the memorypolynomial given by Equation (43): $\begin{matrix}{{\hat{y}(n)} = {\sum\limits_{m = 0}^{m = M}{{x\left( {n - m} \right)}\quad{\sum\limits_{k = 1}^{k = P}{c_{mk}{{x\left( {n - m} \right)}}^{k - 1}}}}}} & (43)\end{matrix}$

The values of the coefficients c_(mk) are given by Equation (44):$\begin{matrix}\begin{matrix}{\left\{ {c_{00},c_{01},c_{02},c_{0k},c_{03},c_{04},c_{05},{\ldots\quad c_{0P}},} \right\},} \\{{{for}\quad{the}\quad{first}\quad{tap}\quad{operating}\quad{over}\quad{x(n)}},} \\{\left\{ {c_{10},c_{11},c_{12},c_{1k},c_{13},c_{14},c_{15},{\ldots\quad c_{1P}},} \right\},} \\{{{for}\quad{the}\quad{second}\quad{tap}\quad{operating}\quad{over}\quad{x\left( {n - 1} \right)}},} \\\vdots \\{\left\{ {c_{M0},c_{M1},c_{M2},c_{Mk},c_{M3},c_{M4},c_{M5},{\ldots\quad c_{{MP},}}} \right\},} \\{{for}\quad{the}\quad{last}\quad{tap}\quad{operating}\quad{over}\quad{{x\left( {n - M} \right)}.}}\end{matrix} & (44)\end{matrix}$

To estimate the number of delays, or the value of M in Equation (43),the number of taps spaced at a single sample or sparsely spaced arevaried while the error signal |e_(inv)(n)|=|y(n)−ŷ(n)| is monitored. Thetap delay at which the error is minimized is taken as the memory delay.The memory delay is passed from off-line model module 52 to inversemodel module 54, which is also used to configure on-line pre-distorter32 that is a replica of module 54. Alternatively, the number of taps Mmay be varied until the total tap power is 10 dB below the magnitude ofthe first tap, which is the linear term of each tap.

Step 214: Estimating and Reducing PA Non-Linearity Effects

Using off-line model module 54, the PA non-linearity may be estimatedand reduced by passing the off-line estimated pre-distorter coefficientsto on-line pre-distorter 32, which is a copy of the off-line inversemodel. The power amplifier non-linearity effects are reduced inaccordance with the memory depth M at step 214. The non-linearityeffects may be reduced by estimating the non-linearity of poweramplifier 40, determining pre-distortion weights to compensate for thenon-linearity in accordance with the memory depth, and applying thepre-distortion weights. Inverse model module 54 may use any suitabletechnique to generate the pre-distorter weights applied to 32. Forexample, indirect learning may be used to train the off-line modelmodule using the least means squares (LMS) or the recursive leastsquares (RLS) error minimization techniques.

According to the recursive least squares minimization technique,estimated signal {circumflex over (x)}_(PD)(n) may be given by Equation(45):{circumflex over (x)} _(PD)(n)=ŵ ^(H)(n−1)y(n)   (45)The inverse modeling error signal e_(inv)(n) measures the differencebetween the pre-distorted signal x_(PD)(n) and estimated signal{circumflex over (x)}_(PD)(n), and may be given by Equation (46):e _(inv)(n)=x _(PD)(n)−{circumflex over (x)} _(PD)(n)   (46)

The weights ŵ(n) of the off-line processor may be iteratively generatedfrom error signal e_(inv)(n) according to Equation (47):ŵ(n)=ŵ(n−1)+K(n)e _(inv)*(n)   (47)where K(n) represents the Kalman gain given by Equation (48):$\begin{matrix}{{K(n)} = \frac{\lambda^{- 1}{P\left( {n - 1} \right)}\quad{x_{PD}(n)}}{1 + {\lambda^{- 1}{x_{PD}^{H}(n)}\quad{P\left( {n - 1} \right)}\quad{x_{PD}(n)}}}} & (48)\end{matrix}$where P(n) represents the inverse correlation matrix given by Equation(49):P(n)=P(n−1)(Iλ ⁻¹−λ⁻¹ K(n)x _(PD) ^(H)(n))   (49)and where constant λ represents the RLS forgetting factor, 0.95≦λ≦1, andI represents the identity matrix. The initialization of the weights andthe inverse correlation matrix may be set by ŵ(0)=0 and P(0)=Iδ, where δis a constant ranging from 10 to 100.

Filter smoother 172 performs a smoothing operation on the weights,Pre-distorter 32 applies the weights to the input signal x(n) to reducethe effects of non-linearity on output signal y(t).

Step 218: Monitoring Performance

The performance of power amplifier 40 is monitored at step 218. Systemmonitor 24 may be used to remotely and continuously monitor the dynamicperformance of power amplifier 40 with information supplied from inverseand forward model modules 52 and 54. System monitor 24 may displaystatistical performance characteristics using any suitable graphicalrepresentation, for example, a graph or a bar chart.

According to one embodiment, system monitor 24 receives input signalx(n), pre-distorted signal x_(PD)(n), and output signal y(n). The PAbaseband input/output before linearization is given by Equation (50):$\begin{matrix}{\frac{y(n)}{x(n)} = \frac{G_{PD}G_{IF}G_{PA}H_{PA}}{G_{fb}}} & (50)\end{matrix}$which represents the PA final AM/AM and AM/PM after linearization. Thepre-distorter linear gain G_(PD) is equal to the initial forward gainsetting that is equal to a₀₁.

The PA baseband input/output after linearization is given by Equation(51): $\begin{matrix}{\frac{y(n)}{x(n)} = \frac{G_{PD}H_{PD}G_{IF}G_{PA}H_{PA}}{G_{fb}}} & (51)\end{matrix}$where the on-line pre-distorter 32 is programmed to be a replica of theinverse transfer function of power amplifier 40${H_{PD} = \frac{1}{H_{PA}}},$and the baseband input/output after linearization is given by$\frac{y(n)}{x(n)} = {\frac{G_{PD}G_{IF}G_{PA}}{G_{fb}}.}$

The pre-distorter forward transfer function, that is the pre-distorterAM/AM and AM/PM, is given by Equation (52): $\begin{matrix}{\frac{x(n)}{x_{PD}(n)} = {G_{PD}H_{PD}}} & (52)\end{matrix}$where G_(PA) represents the PA gain, G_(IF) represents the gain from thebaseband to radio frequency conversion, and G_(fb) represents the totalfeedback gain that includes gain from the conversion from the PA outputcoupler to the input of inverse model module 54. The shape of thepre-distorter transfer function $\frac{x(n)}{x_{PD}(n)}$should be the inverse in shape and sign to the PA transfer function$\frac{y(n)}{x_{PD}(n)} = {\frac{G_{IF}G_{PA}H_{PA}}{G_{fb}}.}$System monitor 24 can supply the shape of these transfer functions toindicate the progress and effectiveness of the PA linearization.

According to one embodiment, the PA gain variation may be determinedfrom signals x(n) and y(n) and estimated linear term x(n)a₀₁ generatedby forward model module 52. The gain variation may be displayed incontinuous time or in blocks of time. In addition, the average outputpower of power amplifier 40 and the peak-to-average power ratio may beestimated. For example, the normalized estimated average output power{circumflex over (P)}_(avg) may be defined by Equation (53):$\begin{matrix}{{\hat{P}}_{avg} = {\frac{1}{N}\quad{\sum\limits_{n = 1}^{N}{\frac{y(n)}{G_{fb}}}^{2}}}} & (53)\end{matrix}$

As another example, the average output power may be estimated from theRaleigh distribution of the magnitude of the output baseband samples ofsignal y(n), where the probability distribution may be given by Equation(54): $\begin{matrix}{{f(r)}_{y}\quad\frac{r}{\sigma_{y}^{2}}\quad{\mathbb{e}}^{- \frac{r^{2}}{2\quad\sigma_{y}^{2}}}} & (54)\end{matrix}$where r=|y(n)| and $\sigma = {\sqrt{\frac{{\hat{P}}_{avg}}{2}}.}$As another example, the output peak power can be estimated from blocksof consecutive buffers of samples of signal y(n).

System monitor 24 may also be used to monitor linearization of poweramplifier 40 from the shape of the pre-distorter transfer function${\frac{x_{PD}(n)}{x(n)} = {{G_{PD}H_{PD}} = {{G_{PD}H_{off\_ i}} = {G_{PD}\frac{1}{H_{PA}}}}}},$and the PA transfer functions before linearization is given by Equation(55): $\begin{matrix}{\frac{y(n)}{x(n)} = \frac{G_{PD}G_{IF}G_{PA}}{G_{fb}}} & (55)\end{matrix}$After linearization, with ${H_{PD} = \frac{1}{H_{PA}}},$the PA AM/AM and AM/FM, which is the forward complex transfer function,is given by$\frac{y(n)}{x(n)} = {\frac{G_{PD}H_{PD}G_{IF}G_{PA}}{G_{fb}}.}$

According to one embodiment, initial samples of signals x(n), x_(PD)(n),and y(n) may be collected to generate and display the inverse andforward AM/AM and AM/PM. System monitor 24 may be used to check whetherthe estimated inverse ${AM}\text{/}{AM}\quad{\frac{x_{PD}(n)}{x(n)}}$is sufficiently close to inverse of the PA AM/AM given by$\frac{y(n)}{x_{PD}(n)}$and that the phase of the pre-distorter transfer function is theopposite (conjugate) of the PA transfer function as expressed byEquation (56): $\begin{matrix}{\tan^{- 1}\left( \frac{x_{PD}(n)}{x_{PD}(n)} \right)} & (56)\end{matrix}$

Alterations or permutations such as modifications, additions, oromissions may be made to the method without departing from the scope ofthe invention. The method may include more, fewer, or other steps.Additionally, steps may be performed in any suitable order withoutdeparting from the scope of the invention.

Certain embodiments of the invention may provide one or more technicaladvantages. A technical advantage of one embodiment may be that off-linecomponents for determining non-linearity of on-line components may belocated remote from the on-line components. Locating the off-linecomponents remote from the on-line components may allow for remotelinearization of the on-line components while reducing power and spacerequirements for the on-line components.

While this disclosure has been described in terms of certain embodimentsand generally associated methods, alterations and permutations of theembodiments and methods will be apparent to those skilled in the art.Accordingly, the above description of example embodiments does notdefine or constrain this disclosure. Other changes, substitutions, andalterations are also possible without departing from the spirit andscope of this disclosure, as defined by the following claims.

1. A method for estimating non-linearity of a power amplifier,comprising: receiving a plurality of signals at a first location, theplurality of signals comprising an input signal, a pre-distorted signal,and an output signal, the input signal being pre-distorted to yield thepre-distorted signal, the pre-distorted signal being amplified by apower amplifier to yield the output signal, the output signal exhibitingdistortion with respect to the input signal, the distortion comprising anon-linearity effect, the power amplifier located at a second locationremote from the first location; estimating non-linearity of the poweramplifier in accordance with the signals at the first location using aninverse model; calculating pre-distortion information according to theestimated non-linearity using the inverse model; and sending thepre-distortion information to a pre-distorter at the second location,the pre-distorter operable to reduce the non-linearity effect using thepre-distortion information.
 2. The method of claim 1, further comprisingreducing an impairment effect of the distortion by: isolating theimpairment effect of the distortion using the inverse model; andcalculating compensation information according to the isolatedimpairment effect; and sending the compensation information to animpairment compensator at the second location, the impairmentcompensator operable to reduce the impairment effect using thecompensation information.
 3. The method of claim 1, wherein estimatingthe non-linearity of the power amplifier further comprises: estimating amemory depth of the power amplifier at the first location using aforward model located at the second location to generate memoryinformation; and estimating at the second location the non-linearity ofthe power amplifier in accordance with the memory information.
 4. Themethod of claim 3, wherein estimating the memory depth of the poweramplifier using further comprises: establishing an output signalpolynomial corresponding to the output signal; establishing a polynomialorder of the output signal polynomial to fit the distortion; anddetermining a number of delay taps to be implemented at thepre-distorter using the output signal polynomial with the establishedpolynomial order.
 5. The method of claim 3, wherein estimating thememory depth of the power amplifier further comprises: establishing anoutput signal polynomial corresponding to the output signal, the outputsignal polynomial comprising a plurality of coefficients; varying anumber of delay taps of the power amplifier; monitoring the plurality ofcoefficients in response to the variation with respect to a memorythreshold; determining that the plurality of coefficients fall below thememory threshold with a certain number of delay taps; and identifyingthe certain number of delay taps as a number of delay taps for modelingthe power amplifier.
 6. The method of claim 1, further comprising:calculating a performance characteristic from the signals, theperformance characteristic describing performance of the poweramplifier; and displaying a graphical representation of the performancecharacteristic using a system monitor at the first location.
 7. Themethod of claim 1, wherein the first location is more than one yard fromthe second location.
 8. The method of claim 1, wherein the firstlocation is more than one mile from the second location.
 9. The methodof claim 1, wherein the first location is more than one hundred milesfrom the second location.
 10. A system for estimating non-linearity of apower amplifier, comprising: an inverse model module at a first locationoperable to: receive a plurality of signals, the plurality of signalscomprising an input signal, a pre-distorted signal, and an outputsignal, the input signal being pre-distorted to yield the pre-distortedsignal, the pre-distorted signal being amplified by a power amplifier toyield the output signal, the output signal exhibiting distortion withrespect to the input signal, the distortion comprising a non-linearityeffect, the power amplifier located at a second location remote from thefirst location; estimate non-linearity of the power amplifier inaccordance with the signals; and calculate pre-distortion informationaccording to the estimated non-linearity; and a transmitter coupled tothe inverse model module and operable to send the pre-distortioninformation to a pre-distorter at the second location, the pre-distorteroperable to reduce the non-linearity effect using the pre-distortioninformation.
 11. The system of claim 10, wherein: the inverse modelmodule is further operable to reduce an impairment effect of thedistortion by: isolating the impairment effect of the distortion; andcalculating compensation information according to the isolatedimpairment effect; and the transmitter is further operable to send thecompensation information to an impairment compensator at the secondlocation, the impairment compensator operable to reduce the impairmenteffect using the compensation information.
 12. The system of claim 10,further comprising a forward model module located at the second locationand operable to estimate a memory depth of the power amplifier togenerate memory information, the inverse model module being operable toestimate the non-linearity of the power amplifier in accordance with thememory information.
 13. The system of claim 12, wherein forward modelmodule is further operable to estimate the memory depth of the poweramplifier by: establishing an output signal polynomial corresponding tothe output signal; establishing a polynomial order of the output signalpolynomial to fit the distortion; and determining a number of delay tapsto be implemented at the pre-distorter using the output signalpolynomial with the established polynomial order.
 14. The system ofclaim 12, wherein forward model module is further operable to estimatethe memory depth of the power amplifier by: establishing an outputsignal polynomial corresponding to the output signal, the output signalpolynomial comprising a plurality of coefficients; varying a number ofdelay taps of the power amplifier; monitoring the plurality ofcoefficients in response to the variation with respect to a memorythreshold; determining that the plurality of coefficients fall below thememory threshold with a certain number of delay taps; and identifyingthe certain number of delay taps as a number of delay taps for modelingthe power amplifier.
 15. The system of claim 10, further comprising asystem monitor at the first location operable to: calculate aperformance characteristic from the signals, the performancecharacteristic describing performance of the power amplifier; anddisplay a graphical representation of the performance characteristic.16. The system of claim 10, wherein the first location is more than oneyard from the second location.
 17. The system of claim 10, wherein thefirst location is more than one mile from the second location.
 18. Thesystem of claim 10, wherein the first location is more than one hundredmiles from the second location.
 19. A system for estimatingnon-linearity of a power amplifier, comprising: means for receiving aplurality of signals at a first location, the plurality of signalscomprising an input signal, a pre-distorted signal, and an outputsignal, the input signal being pre-distorted to yield the pre-distortedsignal, the pre-distorted signal being amplified by a power amplifier toyield the output signal, the output signal exhibiting distortion withrespect to the input signal, the distortion comprising a non-linearityeffect, the power amplifier located at a second location remote from thefirst location; means for estimating non-linearity of the poweramplifier in accordance with the signals at the first location using aninverse model; means for calculating pre-distortion informationaccording to the estimated non-linearity using the inverse model; andmeans for sending the pre-distortion information to a pre-distorter atthe second location, the pre-distorter operable to reduce thenon-linearity effect using the pre-distortion information.
 20. A methodfor estimating non-linearity of a power amplifier, comprising: receivinga plurality of signals at a first location, the plurality of signalscomprising an input signal, a pre-distorted signal, and an outputsignal, the input signal being pre-distorted to yield the pre-distortedsignal, the pre-distorted signal being amplified by a power amplifier toyield the output signal, the output signal exhibiting distortion withrespect to the input signal, the distortion comprising a non-linearityeffect, the power amplifier located at a second location remote from thefirst location, the first location being more than one hundred milesfrom the second location; reducing an impairment effect of thedistortion by: isolating the impairment effect of the distortion usingan inverse model at the first location; and calculating compensationinformation according to the isolated impairment effect; and sending thecompensation information to an impairment compensator at the secondlocation, the impairment compensator operable to reduce the impairmenteffect using the compensation information; estimating at the secondlocation non-linearity of the power amplifier in accordance with thesignals at the first location using the inverse model, the non-linearityestimated by: estimating a memory depth of the power amplifier at thefirst location using a forward model located at the second location togenerate memory information; and estimating the non-linearity of thepower amplifier in accordance with the memory information by:establishing an output signal polynomial corresponding to the outputsignal; establishing a polynomial order of the output signal polynomialto fit the distortion; and determining a number of delay taps to beimplemented at the pre-distorter using the output signal polynomial withthe established polynomial order by: establishing an output signalpolynomial corresponding to the output signal, the output signalpolynomial comprising a plurality of coefficients; varying a number ofdelay taps of the power amplifier; monitoring the plurality ofcoefficients in response to the variation with respect to a memorythreshold; determining that the plurality of coefficients fall below thememory threshold with a certain number of delay taps; and identifyingthe certain number of delay taps as a number of delay taps for modelingthe power amplifier; calculating pre-distortion information according tothe estimated non-linearity using the inverse model; and sending thepre-distortion information to a pre-distorter at the second location,the pre-distorter operable to reduce the non-linearity effect using thepre-distortion information; calculating a performance characteristicfrom the signals, the performance characteristic describing performanceof the power amplifier; and displaying a graphical representation of theperformance characteristic using a system monitor at the first location.